TY - JOUR

T1 - Topological embeddings into random 2-complexes

AU - Farber, Michael

AU - Nowik, Tahl

N1 - Publisher Copyright:
© 2021 Wiley Periodicals LLC.

PY - 2021/7

Y1 - 2021/7

N2 - We consider 2-dimensional random simplicial complexes Y in the multiparameter model. We establish the multiparameter threshold for the property that every 2-dimensional simplicial complex S admits a topological embedding into Y asymptotically almost surely. Namely, if in the procedure of the multiparameter model on n vertices, each i-dimensional simplex is taken with probability pi = pi(n), then the threshold is (Formula presented.). Our claim in one direction is in fact slightly stronger, namely, we show that if (Formula presented.) is sufficiently larger than (Formula presented.) then every S has a fixed subdivision S′ which admits a simplicial embedding into Y asymptotically almost surely. In the other direction we show that if (Formula presented.) is sufficiently smaller than (Formula presented.), then asymptotically almost surely, the torus does not admit a topological embedding into Y.

AB - We consider 2-dimensional random simplicial complexes Y in the multiparameter model. We establish the multiparameter threshold for the property that every 2-dimensional simplicial complex S admits a topological embedding into Y asymptotically almost surely. Namely, if in the procedure of the multiparameter model on n vertices, each i-dimensional simplex is taken with probability pi = pi(n), then the threshold is (Formula presented.). Our claim in one direction is in fact slightly stronger, namely, we show that if (Formula presented.) is sufficiently larger than (Formula presented.) then every S has a fixed subdivision S′ which admits a simplicial embedding into Y asymptotically almost surely. In the other direction we show that if (Formula presented.) is sufficiently smaller than (Formula presented.), then asymptotically almost surely, the torus does not admit a topological embedding into Y.

KW - multiparameter threshold

KW - random simplicial complexes

UR - http://www.scopus.com/inward/record.url?scp=85099062420&partnerID=8YFLogxK

U2 - 10.1002/rsa.20987

DO - 10.1002/rsa.20987

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AN - SCOPUS:85099062420

SN - 1042-9832

VL - 58

SP - 664

EP - 675

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

IS - 4

ER -